3.73 \(\int f^{a+b x^2} x^5 \, dx\)

Optimal. Leaf size=62 \[ -\frac{x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac{f^{a+b x^2}}{b^3 \log ^3(f)}+\frac{x^4 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

f^(a + b*x^2)/(b^3*Log[f]^3) - (f^(a + b*x^2)*x^2)/(b^2*Log[f]^2) + (f^(a + b*x^2)*x^4)/(2*b*Log[f])

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Rubi [A]  time = 0.0628613, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ -\frac{x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac{f^{a+b x^2}}{b^3 \log ^3(f)}+\frac{x^4 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^5,x]

[Out]

f^(a + b*x^2)/(b^3*Log[f]^3) - (f^(a + b*x^2)*x^2)/(b^2*Log[f]^2) + (f^(a + b*x^2)*x^4)/(2*b*Log[f])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{a+b x^2} x^5 \, dx &=\frac{f^{a+b x^2} x^4}{2 b \log (f)}-\frac{2 \int f^{a+b x^2} x^3 \, dx}{b \log (f)}\\ &=-\frac{f^{a+b x^2} x^2}{b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^4}{2 b \log (f)}+\frac{2 \int f^{a+b x^2} x \, dx}{b^2 \log ^2(f)}\\ &=\frac{f^{a+b x^2}}{b^3 \log ^3(f)}-\frac{f^{a+b x^2} x^2}{b^2 \log ^2(f)}+\frac{f^{a+b x^2} x^4}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0084153, size = 41, normalized size = 0.66 \[ \frac{f^{a+b x^2} \left (b^2 x^4 \log ^2(f)-2 b x^2 \log (f)+2\right )}{2 b^3 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^5,x]

[Out]

(f^(a + b*x^2)*(2 - 2*b*x^2*Log[f] + b^2*x^4*Log[f]^2))/(2*b^3*Log[f]^3)

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Maple [A]  time = 0.007, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ({b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}-2\,b{x}^{2}\ln \left ( f \right ) +2 \right ){f}^{b{x}^{2}+a}}{2\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)*x^5,x)

[Out]

1/2*(b^2*x^4*ln(f)^2-2*b*x^2*ln(f)+2)*f^(b*x^2+a)/ln(f)^3/b^3

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Maxima [A]  time = 1.16652, size = 63, normalized size = 1.02 \begin{align*} \frac{{\left (b^{2} f^{a} x^{4} \log \left (f\right )^{2} - 2 \, b f^{a} x^{2} \log \left (f\right ) + 2 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^5,x, algorithm="maxima")

[Out]

1/2*(b^2*f^a*x^4*log(f)^2 - 2*b*f^a*x^2*log(f) + 2*f^a)*f^(b*x^2)/(b^3*log(f)^3)

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Fricas [A]  time = 1.53527, size = 100, normalized size = 1.61 \begin{align*} \frac{{\left (b^{2} x^{4} \log \left (f\right )^{2} - 2 \, b x^{2} \log \left (f\right ) + 2\right )} f^{b x^{2} + a}}{2 \, b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^5,x, algorithm="fricas")

[Out]

1/2*(b^2*x^4*log(f)^2 - 2*b*x^2*log(f) + 2)*f^(b*x^2 + a)/(b^3*log(f)^3)

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Sympy [A]  time = 0.124658, size = 54, normalized size = 0.87 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}} \left (b^{2} x^{4} \log{\left (f \right )}^{2} - 2 b x^{2} \log{\left (f \right )} + 2\right )}{2 b^{3} \log{\left (f \right )}^{3}} & \text{for}\: 2 b^{3} \log{\left (f \right )}^{3} \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x**5,x)

[Out]

Piecewise((f**(a + b*x**2)*(b**2*x**4*log(f)**2 - 2*b*x**2*log(f) + 2)/(2*b**3*log(f)**3), Ne(2*b**3*log(f)**3
, 0)), (x**6/6, True))

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Giac [A]  time = 1.27472, size = 58, normalized size = 0.94 \begin{align*} \frac{{\left (b^{2} x^{4} \log \left (f\right )^{2} - 2 \, b x^{2} \log \left (f\right ) + 2\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^5,x, algorithm="giac")

[Out]

1/2*(b^2*x^4*log(f)^2 - 2*b*x^2*log(f) + 2)*e^(b*x^2*log(f) + a*log(f))/(b^3*log(f)^3)